We examine the algebraic structure of closure, semiprime and prime operations on submonoids of natural numbers. We find that the closure operations under composition do not form a submonoid under composition. We also describe all the semiprime operations on natural numbers and show that they are a submonoid.
We investigate the relations among the semiprime operations on ideals of the sub- semi-group (2, 3) and define which of these operations may form a monoid or a left act under composition.
We also consider the algebraic structure of monoids with multiple maximal ideals and generalize these results to higher dimensions.